In a turgid game of few clear chances the Hammers' dead-ball threat troubled their hosts all afternoon, with Angel Rangel hooking a long Razvan Rat free-kick off the line, while Guy Demel and Kevin Nolan also spurned sights of goal.
Jussi Jaaskelainen made two point-blank stops to deny Chico Flores and Wilfried Bony at the other end, but the hosts were most aggrieved by referee Phil Dowd's decision to turn down a stoppage-time appeal for handball against Joe Cole.
The West Ham substitute had his arm raised as he leapt to block a Nathan Dyer cross but the game ended goalless.
Allardyce again opted to start with a side containing no recognised striker, and their 4-6-0 formation stymied Swansea's passing game, while they carried a potent threat from set-pieces at the other end.
Afterwards, Allardyce said: "That's five clean sheets we've got now. Away from home we've only conceded one goal, from a very dubious penalty, and that is going to bring us a lot of points because eventually we will score.
"We obviously want to score more if we can but it looks as though we have to create six or seven chances for ourselves.
"That has got to get better."